Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models

TL;DR

This paper constructs new boundary quantum field theory models using conformal nets associated with lattices and loop groups, expanding the class of known models with explicit examples.

Contribution

It introduces a novel method to generate boundary QFT models from lattice and loop group conformal nets via a recent semigroup construction.

Findings

Explicit elements of the semigroup for lattice conformal nets

Construction of boundary models from level 1 lattice nets

Extension to Spin(n) loop group nets at level 2

Abstract

In this article we give new examples of models in boundary quantum field theory, i.e. local time-translation covariant nets of von Neumann algebras, using a recent construction of Longo and Witten, which uses a local conformal net A on the real line together with an element of a unitary semigroup associated with A. Namely, we compute elements of this semigroup coming from H\"older continuous symmetric inner functions for a family of (completely rational) conformal nets which can be obtained by starting with nets of real subspaces, passing to its second quantization nets and taking local extensions of the former. This family is precisely the family of conformal nets associated with lattices, which as we show contains as a special case the level 1 loop group nets of simply connected, simply laced groups. Further examples come from the loop group net of Spin(n) at level 2 using the orbifold construction.